- #26

samalkhaiat

Science Advisor

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pellman said:what I need then, I guess, is the Maxwell equations (got 'em) + the continuity equation (got it) + plus the continuous version of the Lorentz force in terms of [tex]\rho[/tex] and [tex]J[/tex] and depending on E and B (need it).

If someone could provide that last piece, I'd appreciate it.

You can write

[tex]m= \int \mu d^3x[/tex]

[tex]e= \int \rho d^3x[/tex]

in the Lorentz force

[tex] \frac{dP_{i}}{dt} = \int d^3x \mu \frac{dv_{i}}{dt}= \int d^3x [ \rho E_{i} + \rho \epsilon_{ijk} v_{j} H_{k}][/tex]

or

[tex] \mu \frac{dv_{i}}{dt}= \rho E_{i} +\epsilon_{ijk} J_{j} H_{k}[/tex]

In 4-vector, you have

[tex] \mu \frac{dV^{\mu}}{ds}= \rho F^{\mu\nu} V_{\nu}[/tex]

or

[tex] \mu \frac{dV^{\mu}}{dt}= F^{\mu\nu} J_{\nu}[/tex]

This can be turned to conservation statement.

As for the Lagrangian, it is

[tex]\mathcal{L}= - \mu (1-v^2)^{1/2} -A_{\mu} J^{\mu} -\frac{1}{16\pi} F_{\mu\nu}F^{\mu\nu}[/tex]

regards

sam

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